Algebraic Graph Theory Seminar - Chris Godsil

Title: The 600-cell

Abstract:

If d ≥ 5, then in R^d there are exactly three regular polytopes (simple, hypercube, dual hypercube). If d = 3 we have the icosahedron and the dodecahedron in addition. If d = 4, there are again two exceptional regular polytopes, the so-called 120-cell and 600-cell. I will discuss how these claims can be proved, and will present a construction of the 600-cell using quaternions. The 600-cell is interesting because despite being "highly regular", it is not distance regular; moreover it gives rise to a graph on 60 vertices with quantum chromatic number four and chromatic number five.